Mathematics \ Complex Analysis \ Complex Numbers
Complex numbers are a fundamental concept in both mathematics and its applications to the physical sciences and engineering. They extend the idea of one-dimensional real numbers to the two-dimensional complex plane by introducing a new unit called the imaginary unit, denoted as \(i\), where \(i\) is defined by the property \(i^2 = -1\).
Definition and Notation:
A complex number is typically written in the form \(z = a + bi\), where \(a\) and \(b\) are real numbers. In this representation, \(a\) is called the real part of \(z\), denoted by \(\text{Re}(z)\), and \(b\) is called the imaginary part of \(z\), denoted by \(\text{Im}(z)\). Thus, a complex number can be expressed as:
\[ z = a + bi \quad \text{with} \quad a, b \in \mathbb{R} \quad \text{and} \quad i^2 = -1 \]
Complex Plane:
The set of all complex numbers forms the complex plane, also known as the Argand plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part. Each complex number corresponds to a unique point \((a, b)\) in this plane.
Operations on Complex Numbers:
1. Addition and Subtraction: If \(z_1 = a_1 + b_1i\) and \(z_2 = a_2 + b_2i\), then
\[
z_1 + z_2 = (a_1 + a_2) + (b_1 + b_2)i
\]
\[
z_1 - z_2 = (a_1 - a_2) + (b_1 - b_2)i
\]
Multiplication: If \(z_1 = a_1 + b_1i\) and \(z_2 = a_2 + b_2i\), then
\[
z_1 \cdot z_2 = (a_1a_2 - b_1b_2) + (a_1b_2 + a_2b_1)i
\]Division: Given \(z_1 = a_1 + b_1i\) and \(z_2 = a_2 + b_2i \neq 0\), the quotient \(z_1 / z_2\) is found by multiplying the numerator and the denominator by the conjugate of the denominator:
\[
z_1 / z_2 = \frac{a_1 + b_1i}{a_2 + b_2i} \cdot \frac{a_2 - b_2i}{a_2 - b_2i} = \frac{(a_1a_2 + b_1b_2) + (b_1a_2 - a_1b_2)i}{a_2^2 + b_2^2}
\]
Complex Conjugate:
The complex conjugate of \(z = a + bi\) is \( \overline{z} = a - bi\). The magnitude or modulus \( |z| \) of \( z \) is given by
\[
|z| = \sqrt{a^2 + b^2}
\]
Polar Form:
Complex numbers can also be represented in polar form as \( z = r(\cos \theta + i \sin \theta) \), where \( r = |z| \) is the modulus and \( \theta \) is the argument of \( z \), measured as the angle counterclockwise from the positive real axis. Using Euler’s formula, this can be written as:
\[
z = re^{i\theta}
\]
These properties make complex numbers extremely useful in a variety of mathematical and applied contexts, including solving polynomial equations, performing Fourier transforms, and modeling oscillating systems, among others. Their study provides a critical foundation for more advanced topics within complex analysis.